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Example 1

14.2 Double Integration For more information visit http://www.math.umn.edu/~rogness/multivar/doubleintest.html. Example 1.

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Example 1

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  1. 14.2 Double IntegrationFor more information visithttp://www.math.umn.edu/~rogness/multivar/doubleintest.html

  2. Example 1 Approximate the volume of the solid lying between the paraboloid f(x,y) and the square region R given by the plane 0 < x < 1 and 0 < y < 1 use a partion made up of squares having length of side ¼ (this problem is similar to MRAM from Calc AB)

  3. Solution to Ex 1 It is convenient to choose the centers of the subregions as the points to evaluate f(x,y). Each sub region forms a rectangle with length ¼ and width ¼ and the height is the value of the function. The following values of are the centers of the sub regions

  4. Solution to Example 1 cont The volume can be approximated by

  5. Find the volume under this surface between 0<x<2 and 0<y<1.

  6. z We can sketch the graph by putting in the corners where (x=0, y=0), (x=2, y=0), (x=0, y=1), (x=2, y=1). y x

  7. z The volume of the slice is area . thickness y x We could hold x constant and take a slice through the shape. The area of the slice is given by:

  8. z y x We can add up the volumes of the slices by:

  9. with triangular base between the x-axis, x=1 and y=x. y x slice The base does not have to be a rectangle: thickness of slice area of slice Add all slices from 0 to 1.

  10. Example 2

  11. Solution to Example 2

  12. No human investigation can be called real science if it cannot be demonstrated mathematically. -- Leonardo da Vinci Homework p. 997 1,3,13-23 odd (do not need to do approx for problem 3) Teacher: How much is half of 8? Pupil: Up and down or across? Teacher: What do you mean? Pupil: Well, up and down makes a 3 or across the middle leaves a 0!

  13. Figure 14.17

  14. Figure 14.16

  15. Figure 14.15

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